We provide binomial theorem practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on binomial theorem skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.
List of binomial theorem Questions
| Question No | Questions | Class |
|---|---|---|
| 1 | The number of integral terms in the expansion of ( left(5^{1 / 2}+7^{1 / 8}right)^{1024} ) is | 11 |
| 2 | Coefficient of ( boldsymbol{x}^{k},(mathbf{0} leq boldsymbol{k} leq boldsymbol{n}) ) in expansion of ( boldsymbol{P}=mathbf{1}+(mathbf{1}+boldsymbol{x})+(mathbf{1}+ ) ( boldsymbol{x})^{2} ldots ldots+(1+boldsymbol{x})^{n} ) A. ( ^{n} C_{k} ) B. ( ^{n+1} C_{n-k-1} ) c. ( ^{n} C_{n-k} ) D. ( ^{n+1} C_{k+1} ) |
11 |
| 3 | Find the coefficient of ( x^{-7} ) in the expansion of ( left(boldsymbol{a} boldsymbol{x}-frac{mathbf{1}}{boldsymbol{b} boldsymbol{x}^{2}}right)^{11} ) ( begin{array}{ll}text { A. } & ^{11} C_{5}end{array} ) В. ( ^{10} C_{4} ) c. ( ^{11} C_{4} ) D. ( ^{10} C_{5} ) |
11 |
| 4 | Number of irrational terms in the expansion of ( left(5^{frac{1}{6}}+2^{frac{1}{8}}right)^{100} ) A . 96 B. 97 c. 98 D. 99 |
11 |
| 5 | The sum of the coefficient of first 3 terms in the expansion ( left(x-frac{3}{x^{2}}right)^{m} ) in ( 559 . ) Find the term of the expansion containing ( boldsymbol{x}^{mathbf{3}} ) |
11 |
| 6 | If ( A ) and ( B ) are coefficients of ( x^{n} ) in the expansions of ( (1+x)^{2 n} ) and ( (1+x)^{2 n-1} ) respectively, then ( frac{A}{B} ) is equal to A .4 B . 2 ( c .9 ) D. 6 |
11 |
| 7 | The coefficient of three consecutive terms in the expansion of ( (1+a)^{n} ) are in ratio 1: 7: 21 , then find the value of ( boldsymbol{n} ) |
11 |
| 8 | If ( C_{r} ) denotes the binomial coefficient ( ^{n} C_{r} ) then ( (-1) C_{0}^{2}+2 C_{1}^{2}+5 C_{2}^{2}+ ) ( ldots(3 n-1) C_{n}^{2}= ) A ( cdot(3 n-2)^{2 n} C_{n} ) ( ^{text {В }} cdotleft(frac{3 n-2}{2}right)^{2 n} C_{n} ) c. ( (5+3 n)^{2 n} C_{n} ) D ( cdotleft(frac{3 n-5}{2}right)^{2 n} C_{n+1} ) |
11 |
| 9 | Find ( a ) if the ( 17^{t h} ) and ( 18^{t h} ) term of the expanse on ( (2+a)^{50} ) are equal. |
11 |
| 10 | V1V9 VO101010101 011 The coefficient ofx7 in the expansion of (1-x-x2 + x) is [2011] (a) -132 (b) -144 () 132 (d) 144 |
11 |
| 11 | The sum of all the coefficient of those terms in the expansion of ( (a+b+c+d)^{8} ) which contains ( b ) but not ( boldsymbol{c} ) is ( mathbf{A} cdot 6305 ) B ( cdot 4^{8}-3^{8} ) C. Number of ways of forming 8 digit numbers using digits 1,2,3 each number as atleast one 3 D. Number of ways of forming 4 digit numbers using digits 1,2,3 each number as atleast one 3 |
11 |
| 12 | ( frac{C_{0}}{1}+frac{C_{1}}{2}+frac{C_{2}}{3}+ldots ldots+frac{C_{10}}{11}= ) A ( cdot frac{2^{11}}{11} ) B. ( frac{2^{11}-1}{11} ) c. ( frac{3^{11}}{11} ) D. ( frac{3^{11}-1}{11} ) |
11 |
| 13 | (a) 4 (b) 120 25. If the coefficents of x3 and x4 in the expansion of powers of x are both zero, then (a, b) is equal to: [JEE M 2014] (a) (14,272)) (10,272) (c) (16,251) (a) (14,251) |
11 |
| 14 | Number of rational term is the expansion of ( left(7^{1 / 3}+11^{1 / 9}right)^{729} ) ( A cdot 81 ) B. 82 c. 730 D. None of these |
11 |
| 15 | If ( (1+x)^{n}=sum_{i=0}^{n} C_{i} x^{i}, ) then the sum of the products of ( C_{i} ) ‘s taken two at a time is represented by ( sum_{0 leq i leq j leq n} C_{i} C_{j} ) A ( cdot 2^{n}-frac{(2 n) !}{2(n !)^{2}} ) B. ( 2^{n}+frac{(2 n) !}{2(n !)^{2}} ) c. ( frac{1}{2}left(2^{2 n}+frac{(2 n) !}{(n !)^{2}}right) ) D. ( frac{2^{2 n}}{2(n !)^{2}} ) |
11 |
| 16 | The middle term in the expansion of ( left(frac{a}{x}+b xright)^{12} ) A ( cdot 924 a^{6} b^{6} ) B. ( 924 a^{6} b^{5} ) ( mathbf{c} cdot 924 a^{5} b^{5} ) D. ( 924 a^{5} b^{6} ) |
11 |
| 17 | The number of rational terms in the expansion of ( left(mathbf{9}^{1 / 4}+mathbf{8}^{1 / 6}right)^{1000} ) is: A . 500 в. 400 ( c .501 ) D. none of the above |
11 |
| 18 | A positive integer which is just greater ( operatorname{than}(1+0.0001)^{10000} ) is ( A cdot 3 ) B. 4 ( c .5 ) D. 6 |
11 |
| 19 | et n be positive integer. If the coefficients of 2nd, 3rd, and 1th terms in the expansion of (1 + x)” are in A.P., then the value of n is ………… (1994 – 2 Marks) |
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| 20 | Find ( A_{2}^{n}, ) if the fifth term of the expansion of ( left(sqrt[3]{x}+frac{1}{x}right)^{n} ) does not depend on ( boldsymbol{x} ) |
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| 21 | The coefficient of the middle term in the expansion of ( (1+x)^{2 n} ) is This question has multiple correct options A ( cdot 2^{n} C_{n} ) в. ( frac{1.3 .5 ldots ldots(2 n-1)}{n !} 2^{text {। }} ) c. ( 2.6 ldots(4 n-2) ) D ( cdot 2.4 ldots ldots . .2 n ) |
11 |
| 22 | The Coefficient of ( x^{n} ) in the expansion of ( (1+x)(1-x)^{n} ) is A. ( (n-1) ) B ( cdot(-1)^{n-1} n ) C ( cdot(-1)^{n-1}(n-1)^{2} ) D・ ( (-1)^{n}(1-n) ) |
11 |
| 23 | Find ( (boldsymbol{a}+boldsymbol{b})^{4}-(boldsymbol{a}-boldsymbol{b})^{4} ). Hence, evaluate ( (sqrt{mathbf{3}}+sqrt{mathbf{2}})^{4}-(sqrt{mathbf{3}}-sqrt{mathbf{2}})^{4} ) |
11 |
| 24 | If ( left(boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}+mathbf{1}right)^{boldsymbol{6}}=boldsymbol{a}_{boldsymbol{0}}+left(boldsymbol{a}_{boldsymbol{1}} boldsymbol{x}+frac{boldsymbol{b}_{boldsymbol{1}}}{boldsymbol{x}}right)+ ) ( left(a_{2} x^{2}+frac{b_{2}}{x^{2}}right)+ldots+left(a_{6} x^{6}+frac{b_{6}}{x^{6}}right) ) the find the value of ( a_{0} ) |
11 |
| 25 | The coefficient of ( x^{3} ) in ( left(sqrt{x^{5}}+frac{3}{sqrt{x^{3}}}right)^{6} ) is ( mathbf{A} cdot mathbf{0} ) в. 120 ( c cdot 420 ) D. 540 |
11 |
| 26 | Which term in the expansion of ( (1+x)^{p} cdotleft(1+frac{1}{x}right)^{q} ) is independent of ( x ) where ( p, q ) are positive integers? What is the value of that term? |
11 |
| 27 | Find the coefficient of ( x^{-17} ) in the expansion of ( left(x^{4}-frac{1}{x^{3}}right)^{15} ) A. 1200 B. -1331 c. -1365 D. -2016 |
11 |
| 28 | Expand the following expression in ascending powers of ( x ) as far as ( x^{3} ) ( frac{1+2 x}{1-x-x^{2}} ) |
11 |
| 29 | ( (1+x)^{21}+(1+x)^{22}+ldots+(1+x)^{30} ) coefficient of ( boldsymbol{x}^{5} ) |
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| 30 | The term independent of ( x ) in the expansion of ( left(sqrt{frac{x}{3}}+frac{3}{2 x^{2}}right)^{10} ) will be ( ^{A} cdot frac{3}{2} ) в. ( c cdot frac{5}{2} ) D. None of these |
11 |
| 31 | 13. If the coefficient of x’ equals the (bx) in coefficient of x- the relation (a) a-b=1 , then a and b satisfy [2005] (b) a+b=1 (d) ab=1 |
11 |
| 32 | 26. The sum of coefficients of integral power of x in the binomial expansion (1-27x) is: [JEE M 2015 (b) (250 +1) |
11 |
| 33 | If the constant term in the binomial expansion of ( left(x^{2}-frac{1}{x}right)^{n}, n quad epsilon quad N ) is 15 then the value of ( n ) is equal to |
11 |
| 34 | In the binomial expansion of ( (a- ) ( b)^{n}, n geq 5, ) the sum of 5 th and 6 th terms is zero then a/b equal to A ( cdot frac{5}{n-4} ) B. ( frac{6}{n-5} ) c. ( frac{n-5}{6} ) D. ( frac{n-4}{5} ) |
11 |
| 35 | Expand (i) ( (sqrt{3}+sqrt{2})^{4} ) |
11 |
| 36 | Binomial expansion of ( left(boldsymbol{x}^{k}+frac{mathbf{1}}{mathbf{2}^{mathbf{2 k}}}right)^{mathbf{3 n}} ) where ( n ) is a positive integer, always contains a term which is independent of ( mathbf{A} cdot x^{2} ) B. ( x ) ( mathbf{c} cdot x^{3} ) D. none of the above |
11 |
| 37 | Expand the following binomial ( left(1-3 a^{2}right)^{6} ) | 11 |
| 38 | The coefficient of ( x^{4} ) in the expansion of ( left(frac{x}{2}-frac{3}{x^{2}}right)^{10} ) is equal to: |
11 |
| 39 | Write general term of this:- ( 2 xleft(3+2 x^{2}right)^{20} ) |
11 |
| 40 | Solve : ( left(3 x-frac{1}{2 y}right)left(3 x+frac{1}{2 y}right) ) | 11 |
| 41 | The number of terms whose values depends on ( x ) in the expansion of ( left(x^{2}-2+frac{1}{x^{2}}right)^{n} ) is ( mathbf{A} cdot 2 n+1 ) B. ( 2 n ) ( c ) D. none of these |
11 |
| 42 | 10. The coefficient of x^ in expansion of (1 + x) (1 – x)” is (a) (-1)”-In (b) (-1)” (1-n) [2004 (c) (−1)n-1(n-1) (d) (n-1) |
11 |
| 43 | The middle term of ( left(boldsymbol{x}-frac{1}{boldsymbol{x}}right)^{2 n+1} ) is ( mathbf{A} cdot^{2 n+1} C_{n} cdot x ) B. ( 2 n+1 C_{n} ) C ( cdot(-1)^{n} cdot 2^{2+1} C_{n} ) D ( cdot(-1)^{n} cdot^{2 n+1} C_{n} cdot x ) |
11 |
| 44 | Coefficient of ( x^{5} ) in ( left(1+x^{2}right)^{5}(1+x)^{4} ) is A . 60 B. 80 ( c cdot 90 ) D. 100 |
11 |
| 45 | Show that the middle term in the expansion of ( (1+x)^{2 n} ) is ( frac{1.3 .5 ldots .(2 n-1)}{mid underline{n}} 2^{n} x^{n} ) |
11 |
| 46 | The term independent of ( x ) in ( (1+ ) ( x)^{n}left(1+frac{1}{x}right)^{n} ) is A ( cdot C_{0}^{2}+2 cdot C_{1}^{2}+3 cdot C_{2}^{2}+ldots ldots ldots+(n+1) cdot C_{n}^{2} ) B. ( left(C_{0}+C_{1}+C_{2}+ldots ldots . .+C_{n}right)^{2} ) c. ( C_{0}^{2}+C_{1}^{1}+C_{2}^{2}+ldots ldots . .+C_{n}^{2} ) D. None |
11 |
| 47 | Find ( x, ) if it is known that the second term of the expansion of ( left(x+x^{log x}right)^{5} ) is equal to 1000000 |
11 |
| 48 | Find the ( 13^{t h} ) term in the expansion of ( left(9 x-frac{1}{3 sqrt{x}}right)^{18} ) | 11 |
| 49 | Find the value of a given ( mathbf{3}+frac{mathbf{1}}{mathbf{4}}(mathbf{3}+boldsymbol{p})+frac{mathbf{1}}{mathbf{4}^{2}}(mathbf{3}+mathbf{2} boldsymbol{p})+frac{mathbf{1}}{mathbf{4}^{3}}(mathbf{3}+ ) ( mathbf{3} boldsymbol{p})+ldots=? ) |
11 |
| 50 | The term independent of ( x ) in the expansion of ( left(x^{2}+frac{1}{x}right)^{9} ) is ( A ) B. – c. 48 D. 84 |
11 |
| 51 | Expand ( left(x^{2}+frac{3}{x}right)^{4} ) | 11 |
| 52 | The ratio of fifth term from the beginning to the fifth term from the end in the expansion of ( left(sqrt[4]{2}+frac{1}{sqrt[4]{3}}right)^{n} ) is ( sqrt{6}: 1 . ) If ( n=frac{20}{lambda}, ) find the value of ( lambda ) |
11 |
| 53 | Find the coefficient of ( x^{5} ) in the product ( (1+2 x)^{6}(1-x)^{7} ) using binomial theorem. |
11 |
| 54 | 30. If the fourth term in the Binomial expansion of ( = + xlog8x (x>0) is 20×87, then a value of x is: (JEEM 2019-9 April (M) (a) 8 (6) 8 (c) 8 (d) 82 |
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| 55 | The coefficient of ( x^{160} ) in the expansion of ( left(x^{8}+right. ) 1) ( ^{60}left(x^{12}+3 x^{4}+frac{3}{x^{4}}+frac{1}{x^{12}}right)^{-10} ) A. ( ^{30} C_{6} ) B. ( ^{30} C_{5} ) c. divisible by 189 D. divisible by 203 |
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| 56 | In the expansion of ( (x+sqrt{x^{2}-1})^{6}+ ) ( (x-sqrt{x^{2}-1})^{6}, ) the number of terms is A. 7 B. 14 ( c cdot 6 ) D. |
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| 57 | Which term in the expansion of ( left(frac{x}{3}-frac{2}{x^{2}}right)^{10} ) contains ( x^{4} ? ) ( A ) B. 3 ( c cdot 4 ) D. |
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| 58 | Expand using Binomial Theorem ( left(1+frac{x}{2}-frac{2}{x}right)^{4}, x neq 0 ) | 11 |
| 59 | If rth term in the expansion of ( left(x^{2}+frac{1}{x}right)^{12} ) is independent of ( x, ) then ( boldsymbol{r}= ) ( mathbf{A} cdot mathbf{9} ) B. 8 c. 10 D. none of these |
11 |
| 60 | Find the square of the following binomials by using the identity ( (-z+6) ) | 11 |
| 61 | The value of ( frac{18^{3}+7^{3}+3.18 .7 .25}{3^{6}+6.243 .2+15.81 .4+20.27 .8+15.9 .16+6.3 .32+64} ) ( A cdot 4 ) ( B .3 ) ( c cdot 2 ) D. |
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| 62 | The coefficient of ( x^{4} ) in the expansion of ( {sqrt{1+x^{2}}-x}^{-1} ) in ascending powers of ( x, ) when ( |x|<1 ) is ( mathbf{A} cdot mathbf{0} ) в. ( frac{1}{2} ) ( c cdot-frac{1}{2} ) D. ( -frac{1}{8} ) |
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| 63 | The sum of the last eight coefficients in the expansion of ( (1+x)^{15}, ) is ( ? ) A ( cdot 2^{16} ) B . ( 2^{15} ) ( c cdot 2^{14} ) D. none of these |
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| 64 | if the coefficient of the middle term in the expansion of ( (1+x)^{2 n+2} ) and ( p ) and the coefficients of middle terms in the expansion of ( (1+x)^{2 n+1} ) are ( q ) and ( r ) then A ( . p+q=r ) в. ( p+r=q ) c. ( p=q+r ) D. ( p+q+r=0 ) |
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| 65 | If the first three terms in the expansion of ( (1+a x)^{n} ) are ( 1,8 x, 24 x^{2} ) respectively, then ( a= ) ( mathbf{A} cdot mathbf{1} ) B . 2 ( c cdot 4 ) D. |
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| 66 | The coeff. of ( 8^{t h} ) term in the expansion of ( (1+x)^{10} ) is A. 120 B. 7 c. ( ^{10} C_{8} ) D. 210 |
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| 67 | If the coefficient of 4 consecutive terms in the expansion of ( (1+x)^{n} ) are ( a_{1}, a_{2}, a_{3}, a_{4} ) respectively, then show that: ( frac{boldsymbol{a}_{mathbf{1}}}{boldsymbol{a}_{mathbf{1}}+boldsymbol{a}_{mathbf{2}}}+frac{boldsymbol{a}_{mathbf{3}}}{boldsymbol{a}_{mathbf{3}}+boldsymbol{a}_{boldsymbol{4}}}=frac{mathbf{2} boldsymbol{a}_{boldsymbol{3}}}{boldsymbol{a}_{mathbf{2}}+boldsymbol{a}_{boldsymbol{3}}} ) |
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| 68 | 6. mial expansion of (a – b)”, n25, the sum of the 5th and 6th terms is zero. Then alb equals (20015) In the hi (a)(n-5) 16 (0)55-4) (6) [m-415 (d) Gen 5) |
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| 69 | Find the coefficient of ( x^{50} ) in the expression: ( (1+x)^{1000}+2 x(1+x)^{999}+ ) ( mathbf{3} x^{2}(mathbf{1}+boldsymbol{x})^{mathbf{9} 9 mathbf{8}}+ldots .+mathbf{1 0 0} mathbf{1} boldsymbol{x}^{mathbf{1 0 0 0}} ) A ( .^{1000} mathrm{C}_{50} ) В. ( ^{1001} mathrm{C}_{50} ) ( mathbf{c} cdot^{1002} C_{50} ) D. ( ^{1003} Omega_{50} ) |
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| 70 | In the expansion of ( left(sqrt[3]{4}+frac{1}{sqrt[4]{6}}right)^{20} ) This question has multiple correct options A. the number of rational terms ( =4 ) B. the number of irrational terms ( =19 ) C. the middle term is irrational D. the number of irrational terms ( =17 ) |
11 |
| 71 | Expand ( (1-2 x)^{5} ) | 11 |
| 72 | r and n are positive integers r> 1, n > 2 and coefficient of (r+2)th term and 3rth term in the expansion of (1 + x)2n are equal, then n equals [2002] (2) 3r (6) 3r+1 (c) 2r (d) 2r+1 |
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| 73 | If the fourth term in the expansion of ( left(sqrt{frac{1}{x^{log x+1}}}+x^{1 / 12}right)^{6} ) is equal to 200 and ( x>1, ) then ( x ) is equal to A ( cdot 10^{sqrt{2}} ) 2 ( sqrt{2} ) B. 10 ( c cdot 10^{4} ) D. None of these |
11 |
| 74 | Coefficient of ( boldsymbol{x}^{mathbf{5 0}} ) ( (x>0), ) in ( (1+x)^{1000}+2 x(1+ ) ( boldsymbol{x})^{999}+mathbf{3} boldsymbol{x}^{2}(mathbf{1}+boldsymbol{x})^{998}+ldots ) is A. ( 1000 C_{50} ) B. ( ^{1000} C_{50} ) c. ( ^{1002} 250 ) D. ( ^{1000} C_{49} ) |
11 |
| 75 | The coefficient of the middle term in the binomial expansion in power of ( x ) of ( (1+alpha x)^{4} ) and of ( (1-alpha x)^{6} ) is the same if ( alpha ) equals- A ( cdot-frac{5}{3} ) в. ( frac{10}{3} ) ( c cdot frac{-3}{10} ) D. |
11 |
| 76 | In the expansion of the expression ( (x+a)^{15}, ) if the eleventh term in the geometric mean of the eighth and twelfth terms, which term in the expression is the greatest? A. ( T_{6} ) в. ( T_{7} ) c. ( T_{8} ) D. ( T_{9} ) |
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| 77 | The ( 3 r d, 4 t h ) and 5 th terms in the expansion of ( (1+x)^{n} ) are 60,160 and 240 respectively, then ( x= ) ( A cdot 2 ) B. 4 ( c .5 ) D. 6 |
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| 78 | Assertion ( (sqrt{2}-1)^{n} ) can be expressed as ( sqrt{N} ) ( sqrt{N-1} ) for ( forall N>1 ) and ( n in N ) Reason ( (sqrt{2}-1)^{n} ) can be written in the form ( boldsymbol{alpha}+boldsymbol{beta} sqrt{boldsymbol{2}} forall, boldsymbol{alpha}, boldsymbol{beta} ) are integers & n is a positive integer. A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion, B. Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion, c. Assertion is true but Reason is false D. Assertion is false but Reason is true. |
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| 79 | The number of terms which are free from radical signs in the expansion of ( left(y^{frac{1}{5}}+x^{frac{1}{10}}right)^{55} ) are A. 5 B. 6 ( c cdot 7 ) D. none of these |
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| 80 | Find the cube of the following binomial expressions: ( frac{3}{x}-frac{2}{x^{2}} ) |
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| 81 | If ( |x|<1 ) then the coefficient of ( x^{n} ) in expansion of ( left(1+x+x^{2}+x^{3} dotsright)^{2} ) is ( A ) B . ( n-1 ) ( mathbf{c} cdot n+2 ) ( mathbf{D} cdot n+1 ) |
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| 82 | If the constant term of the binomial zpansion ( left(2 x-frac{1}{x}right)^{n} ) is -160 , then ( n ) is equal to A .4 B. 6 ( c cdot 8 ) D. 10 |
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| 83 | The term in dependent of ( x ) in ( left(1+x+2 x^{3}right)left(frac{3 x^{2}}{2}-frac{1}{3 x}right)^{9} ) A ( cdot frac{25}{54} ) в. ( frac{17}{54} ) ( c cdot frac{1}{6} ) D. ( -frac{17}{51} ) |
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| 84 | Find the 7 th term from the end in the expansion of ( left(2 x^{2}-frac{3}{2 x}right)^{8} ) |
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| 85 | If ( n ) is an integer between 0 and 21 , then the minimum value of ( n !(21-n) ! ) is attained for ( n= ) A . B. 10 c. 12 D. 20 |
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| 86 | The sum of the co-efficients of all odd degree terms in the expansion of ( (x+sqrt{x^{3}-1})^{5}+(x+sqrt{x^{3}-1})^{5} ) ( (x>1) ) is: A . 2 B. – ( c cdot 0 ) ( D ) |
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| 87 | Find the ( 13^{t h} ) terms in the expansion of ( left(9 x-frac{1}{3 sqrt{x}}right)^{18}, x neq 0 ) A . 18564 B. 87328 c. 17374 D. 35546 |
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| 88 | Write the general term in the expansion of ( left(x^{2}-y^{2}right)^{6} ) | 11 |
| 89 | The term independent of ( x ) in ( left(frac{1}{2} x^{frac{1}{3}}+right. ) ( left.boldsymbol{x}^{frac{-1}{5}}right)^{8} ) is A. ( frac{35}{8} ) B. 7 c. ( frac{7}{2} ) D . 28 |
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| 90 | Find the middle term(s) in the expansion of : ( left(2 a x-frac{b}{x^{2}}right)^{12} ) A ( cdot frac{59136 a^{6} b^{6}}{x^{6}} ) В. ( frac{59163 a^{5} b^{5}}{x^{5}} ) c. ( frac{59631 a^{7} b^{7}}{x^{7}} ) D. None of these |
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| 91 | Which number is larger (1.1) ( ^{100000} ) or ( mathbf{1 0}, mathbf{0 0 0} ? ) |
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| 92 | If ( a ) and ( b ) are distinct integers, prove that ( a-b ) is a factor of ( a^{n}-b^{n} ) whenever ( n ) is a positive integer. |
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| 93 | Find ( 7^{t h} ) term of ( left(frac{4 x}{5}-frac{5}{2 x}right)^{9} ) A. ( frac{10050}{x^{3}} ) в. ( frac{10500}{x^{3}} ) c. ( frac{1050}{x^{3}} ) D. ( frac{1000}{x^{3}} ) |
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| 94 | Given positive integers ( i>1, n>2 ) and that the coefficients of ( (3 r)^{t h} ) and ( (r+ ) 2) ( ^{t h} ) terms in the bionomial expansion of ( (1+x)^{2 n} ) are equal, then A ( . n=2 r ) B. n=3r c. ( n=2 r+1 ) D. None of these. |
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| 95 | Given positive integers r>1, n >2 and that the coefficient of (3r)th and (r + 2)th terms in the binomial expansion of (1+ x)2n are equal . Then (1983 – 1 Mark) (a) n=2r (c) n=2r+1 (c) n=3r (d) none of these |
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| 96 | The number of dissimilar terms in the expansion of ( left(1-3 x+3 x^{2}-x^{3}right)^{20} ) is A . 21 B. 32 c. 41 D. 61 |
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| 97 | The sum of the coefficients in the first three terms of the expansion of ( left(x^{2}-frac{2}{x}right)^{m} ) is equal to ( 97 . ) Find the term of the expansion containing ( boldsymbol{x}^{4} ) |
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| 98 | In the expansion of ( (a+b)^{n}, ) the ratio of the binomial coefficients of ( 2^{n d} ) and ( 3^{r d} ) terms is equal to the ratio of the binomial coefficients of ( 5^{t h} ) and ( 4^{t h} ) terms, then ( n= ) A . 4 B. 5 ( c .6 ) D. |
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| 99 | The term independent of ( x ) in ( (2 x- ) ( left.frac{1}{2 x^{2}}right)^{12} ) is ( mathbf{A} cdot-^{12} C_{3} cdot 2^{6} ) B. ( -^{12} C_{5} .2^{2} ) c. ( 12 C_{6} ) D. ( ^{12} C_{4} .2^{4} ) |
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| 100 | If ( n ) is a positive integer and ( (5 sqrt{5}+ ) ( mathbf{1 1})^{2 n+1}=I+f ) where I is an integer and ( 0<f<1 ) then This question has multiple correct options A. I is an even integer B. ( (I+f)^{2} ) is divisible by ( 2^{2 n+1} ) c. lis divisible by 22 D. None of these |
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| 101 | The coefficient of the term independent of ( x ) in the expansion of ( left(frac{x+1}{x^{frac{2}{3}}-x^{frac{1}{3}}+1}-frac{x-1}{x-x^{frac{1}{2}}}right)^{10} ) ( A cdot 70 ) в. 112 c. 105 D. 210 |
11 |
| 102 | Find the middle term in the expansion of ( left(frac{2 x^{2}}{3}-frac{3}{2 x}right)^{12} ) | 11 |
| 103 | If in the expansion of ( (1-x)^{2 n-1}, ) the coefficient of ( x^{r} ) denoted by ( a_{r}, ) then : ( mathbf{A} cdot a_{r-1}+a_{2 n-r}=0 ) В ( cdot a_{r-1}-a_{2 n-r}=0 ) c. ( a_{r-1}+2 a_{a n-r}=0 ) D. None of these |
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| 104 | If ( (1+x)^{10}=a_{0}+a_{1} x+a_{2} x^{2}+dots+ ) ( a_{10} x^{10}, ) then the value of ( left(a_{0}-a_{2}+a_{4}-a_{6}+a_{8}-a_{10}right)^{2}+ ) ( left(a_{1}-a_{3}+a_{5}-a_{7}+a_{9}right)^{2} ) is ( mathbf{A} cdot 2^{10} ) B . 2 ( c cdot 2^{20} ) D. None of these |
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| 105 | The ( 4 t h ) term from the end in the expansion of ( left(frac{x^{3}}{2}-frac{2}{x^{2}}right)^{7} ) is A . ( 35 x ) B. ( 70 x^{2} ) ( c cdot 35 x^{2} ) D. ( 70 x ) |
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| 106 | The number of rational terms in the expansion of ( left(3 frac{1}{4}+7 frac{1}{6}right)^{144} ) A . 33 B. 23 ( c cdot 12 ) D. 13 |
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| 107 | If the 20 th and 21 st terms in the expansion of ( (1+x)^{40} ) are equal, then the value of ( x ) is A ( cdot frac{20}{21} ) в. ( frac{21}{20} ) c. 25 D. ( frac{1}{25} ) |
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| 108 | Find the term which has the exponent of ( x ) as 8 in the expansion of ( left(x^{frac{5}{2}}-frac{3}{x^{3} sqrt{x}}right)^{10} ) ( mathbf{A} cdot T_{2} ) В. ( T_{3} ) c. ( T_{4} ) D. Does not exist |
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| 109 | For ( boldsymbol{r}=mathbf{0}, mathbf{1}, mathbf{2}, mathbf{3}, dots, mathbf{1 0}, ) let ( boldsymbol{A}_{r}, boldsymbol{B}_{r}, boldsymbol{C}_{boldsymbol{r}} ) denote respectively the coefficient of ( x^{r} ) in the expansions of ( (1+x)^{10},(1+x)^{20} ) and ( (1+x)^{30} . ) Then ( sum_{r=1}^{10} A_{r}left(B_{10} B_{r}-right. ) ( left.C_{10} A_{r}right) ) is equal to ( mathbf{A} cdot B_{10}-C_{10} ) B . ( A_{10}left(B_{10}^{2}right)-C_{10} A_{10} ) c. D. ( C_{10}-B_{10} ) |
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| 110 | Number of irrational terms in the binomial expansion of ( left(3^{1 / 5}+7^{1 / 3}right)^{100} ) is A . 94 B. 88 c. 93 D. 95 |
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| 111 | Show that the coefficient of ( a^{m} ) and ( a^{n} ) in the expansion of ( (1+a)^{m+n} ) are equal. |
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| 112 | Given that the ( 4^{t h} ) term in the expansion of ( left(2+frac{3 x}{8}right)^{10} ) has the maximum numerical value, then ( x ) can lie in the interval(s) This question has multiple correct options ( mathbf{A} cdotleft(2, frac{64}{21}right) ) B ( cdotleft(-frac{60}{23},-2right) ) ( mathbf{C} cdotleft(-frac{64}{21},-2right) ) D ( cdotleft(2,-frac{60}{23}right) ) |
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| 113 | In any binomial expansion, the number of terms are ( A cdot geq 5 ) B. ( geq 2 ) ( c cdot geq 3 ) ( D cdot geq 4 ) |
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| 114 | In the expansion of ( left(boldsymbol{a} sqrt{boldsymbol{a}}+frac{mathbf{1}}{boldsymbol{a}^{4}}right)^{boldsymbol{n}} ), the coefficient in the second term exceeds by 44 the coefficient in the first term. Find ( n ) A . 20 B . 25 ( c .35 ) D. 45 |
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| 115 | If the sum of the coefficients of ( x^{2} ) and coefficients of ( x ) in the expansion of ( (1+x)^{m}(1-x)^{n} ) is equal to ( -m, ) then the value of ( 3(n-m) ) is (Note ( : boldsymbol{m}, boldsymbol{n} text { are distinct }) ) |
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| 116 | Using Binomial theorem, evaluate ( (mathbf{9 9})^{5} ) | 11 |
| 117 | The product of two middle terms in the expansion of ( left(frac{3 x^{2}}{2}-frac{1}{3 x}right)^{9} ) is ( ^{mathrm{A}} cdotleft(^{9} C_{4}right)^{2} cdot frac{x^{9}}{512} ) в. ( -9_{C_{4}} .^{9} C_{5}, frac{x^{8}}{512} ) c. ( _{-9}^{text {g }} q_{4} .^{9} C_{5} ). ( frac{x^{9}}{512} ) D・ ( _{9} C_{4} .^{9} C_{5}, frac{x^{9}}{256} ) |
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| 118 | Solve ( (1+i)^{4}+(1-i)^{4}= ) | 11 |
| 119 | Find the expansion of ( (boldsymbol{a}-boldsymbol{2} boldsymbol{x})^{boldsymbol{7}} ) | 11 |
| 120 | In the expansion of ( left(x^{3}-frac{1}{x^{2}}right)^{n}, n in N ) if the sum of the coefficient of ( x^{5} ) and ( x^{10} ) is ( 0, ) then ( n ) is A . 25 B. 20 c. 15 D. None of these |
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| 121 | Multiply the binomials. ( (y-8) ) and ( (3 y-4) ) |
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| 122 | Find the middle term in the expansion of ( (5 x-7 y)^{7} ) | 11 |
| 123 | 15. If the expansion in powers of x of the function is ao +ajx+azx? +azx?… then a, is (1 – ax)(1-bx) 6″ -ah b-a [2006] (b) a” – 6” b-a bn+1-an+1 b-a an+l – 11+1 b-a |
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| 124 | Let ( n ) be a positive integer such that ( left(1+x+x^{2}right)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+ ) ( ldots+a_{2 n} x^{2 n}, ) then ( a_{r}= ) В ( cdot a_{2 n} ; 0 leq r leq 2 n ) D. None of these |
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| 125 | The expansion ( left[boldsymbol{x}+left(boldsymbol{x}^{mathbf{3}}-mathbf{1}right)^{mathbf{1} / mathbf{2}}right]^{mathbf{5}}+[boldsymbol{x}- ) ( left.left(x^{3}-1right)^{1 / 2}right]^{5} ) is a polynomial of degree A. 5 B. 6 ( c cdot 7 ) ( D ) |
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| 126 | If the sum of the coefficients in the expansion of ( (a+b)^{n} ) is ( 4096, ) then the greatest coefficient in the expansion is A ( cdot 924 ) в. 792 ( c .1594 ) D. None of these |
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| 127 | In the expansion of ( (1+x)^{n}, ) the ( 5^{t h} ) term is 4 times the ( 4^{t h} ) term and the ( 4^{t h} ) term is 6 times the ( 3^{r d} ) term. than ( n= ) ( A cdot 9 ) B. 10 ( c cdot 11 ) D. 12 |
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| 128 | Find the 7 th term from the end in the expansion of ( left(9 x-frac{1}{3 sqrt{x}}right)^{18}, x neq 0 ) |
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| 129 | The number of terms with integral coefficients in the expansion of ( left(7^{1 / 3}+5^{1 / 2} cdot xright)^{600} ) is A. 100 B. 50 ( c .101 ) D. none of these |
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| 130 | The coeffcient of ( x^{10} ) in the expansion of ( (1+x)^{2}left(1+x^{2}right)^{3}left(1+x^{3}right)^{4} ) is equal to A . 52 B. 44 c. 50 D. 56 |
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| 131 | 3. If (itaa)? 1 +880 +2438 t… thena…. andon… | 11 |
| 132 | Find the middle terms of the equation of ( left(x^{4}-frac{1}{x^{3}}right)^{11} ) ( mathbf{A} cdot-462 x^{9}, 462 x^{2} ) B . ( -462 x^{8}, 462 x^{4} ) c. ( 462 x^{7},-462 x^{3} ) D. None of these |
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| 133 | In the expansion of ( (sqrt{2}+sqrt[3]{5})^{20} ) the number of rational terms will be: ( A cdot 3 ) B. 10 ( c cdot 4 ) D. |
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| 134 | Find the coefficient of ( x^{5} ) and ( x^{-15} ) in the expansion of ( left(3 x^{2}-frac{1}{3 x^{3}}right)^{10} ? ) |
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| 135 | The total number of terms in the expansion of ( (x+a)^{100}+(x-a)^{100} ) after simplification is A .202 B. 51 ( c .50 ) D. 49 |
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| 136 | Find the middle term(s) of ( left(frac{x^{3 / 2} y}{2}+right. ) ( left.frac{2}{x y^{3 / 2}}right)^{13} ) |
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| 137 | Expand ( left(x^{2}+2 aright)^{5} ) by binomial theorem. |
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| 138 | Expand the binomial ( left(frac{2 x}{3}+frac{3 y}{2}right)^{20} u p ) to four terms. | 11 |
| 139 | 7. The number of integral terms in the expansion of (13+ 5)256 is [2003] (a) 3 (6) 32 (6) 33 (d) 34 |
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| 140 | For ( mathbf{r}=mathbf{0}, mathbf{1}, dots, 10, ) let ( mathbf{A}_{mathbf{r}}, mathbf{B}_{mathbf{r}} ) and ( mathbf{C}_{mathbf{r}} ) denote, respectively, the coefficient of ( x^{r} ) in the expansions of ( (1+x)^{10},(1+ ) ( mathbf{x})^{20} ) and ( (mathbf{1}+mathbf{x})^{30} ) Then ( sum_{r=1}^{10} boldsymbol{A}_{r}left(boldsymbol{B}_{10} boldsymbol{B}_{r}-right. ) ( left.C_{10} A_{r}right) ) is equal to A. ( mathrm{B}_{10}-mathrm{C}_{10} ) B . ( A_{10}left(B_{10}^{2}-C_{10} A_{10}right) ) c. 0 D. ( C_{10}-B_{10} ) |
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| 141 | Find the middle terms in the expansion of ( (5 x-7 y)^{7} ) |
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| 142 | (0 ) 10 10 Let S = { j (j – 1)!°C,, S, = $ 710c; and S3 = 2,2 10C j=1 j=1 statement-1:S = 55 x 29. Statement-2: S, =90 x 2 [2010] nt-2: S = 90 x 28 and S. = 10 x 28. Statement – 1 is true. Statement -2 is true ; Statement-2 not a correct explanation for Statement-1. Statement -1 is true, Statement -2 is false. Statement – 1 is false, Statement -2 is true. statement – lis true, Statement 2 is true; Statement -2 1 1 . c…datamant – |
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| 143 | If ‘p’ and ‘q’ are the coefficients of ( x^{a} ) and ( x^{b} ) respectively in ( (1+x)^{a+b}, ) then A. ( 2 p=q ) В. ( p+q=0 ) c. ( p=q ) D. ( p=2 q ) |
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| 144 | 11. For r=0, 1, …, 10, let A, B and C, denote, respectively, the coefficient of x’ in the expansions of (1 + x)”, (2010) 10 (1 + x)20 and (1 + x)30. Then ZA(B10B.-C104,) is equal to (a) B10-C10 (b) A10(B216C10410 (d) C10-B10 |
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| 145 | Find the middle term(s) in the expansion of : ( left(3 x-frac{x^{3}}{6}right)^{9} ) A ( cdot frac{189}{8} x^{15},-frac{21}{16} x^{17} ) В ( cdot frac{189}{8} x^{17},-frac{21}{16} x^{19} ) C. ( frac{189}{7} x^{15},-frac{23}{13} x^{19} ) D. None of thes |
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| 146 | The coefficient of ( boldsymbol{x}^{r}[mathbf{0} leq boldsymbol{r} leq boldsymbol{n}-mathbf{1}] ) in the expression of ( (x+2)^{n-1}+(x+ ) 2) ( ^{n-2} cdot(x+1)+(x+2)^{n-3} cdot(x+1)^{2}+ ) ( ldots+(x+1)^{n-1} ) is A ( cdot^{n} C_{r}left(2^{r}-1right) ) B. ( ^{n} C_{r}left(2^{n-r}-1right) ) c. ( ^{n} C_{r}left(2^{r}+1right) ) D. ( ^{n} C_{r}left(2^{n-r}+1right) ) |
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| 147 | la J in the expansion of (1 + x)” (1 – x)”, the coefficients of x od r2 are 3 and – 6 respectively, then mis (1999 – 2 Marks) a) 6 (6) 9 (c) 12 (d) 24 |
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| 148 | 16. b-a For natural numbers m, nif (1-y)” (1 + y)” =1+ay +a,y2 + ……. and a, = a, = 10, then (m, n) is (a) (20,45) (b) (35,20) [2006] (c) (45,35) (d) (35,45) ceth |
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| 149 | Find the middle term of ( left(frac{a}{x}+frac{x}{a}right)^{10} ) | 11 |
| 150 | The coefficient of ( x^{30} ) in the expansion of ( left(1+2 x+3 x^{2}+dots .21 x^{20}right)^{2} ) is A . 2706 в. 2450 ( c .1481 ) D. 256 |
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| 151 | If it is known that the third term of the binomial expansion ( left(x+x^{log _{10} x}right)^{3} ) is ( 10^{6} ) then ( x ) is equal to A . 10 B. ( 10^{frac{5}{2}} ) c. 100 D. 5 |
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| 152 | The coefficient of ( x ) in ( left(x^{2}+frac{c}{x}right)^{5} ) is A . 20 B. 10 ( c cdot 10 c^{3} ) D. 20 ( c^{3} ) |
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| 153 | State the whether given statement is true or false Prove that the coefficient of xnxn A. True B. False |
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| 154 | 28. The value of (21C, -10C,)+(°C, – 10C,)+(!Cz – 1°C3)+(1C4 – 10C) +…+(+1C70-10C10) is: [JEE M 2017 (a) 220 -210 (b) 221 – 211 (c) 221 – 210 (d) 220 – 29 11111 |
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| 155 | If in the expansion of ( left(frac{1}{x}+x tan xright)^{5}, ) the ratio of ( 4^{t h} ) term to the ( 2^{n d} ) term is ( frac{2}{27} pi^{4} ) then the value of ( x ) can be A ( cdot frac{-pi}{6} ) в. ( frac{-pi}{3} ) c. D. ( frac{pi}{12} ) |
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| 156 | If the coefficients of ( a^{m} ) and ( a^{n} ) in the expansion of ( (1+a)^{m+n} ) are ( alpha ) and ( beta ) then which one of the following is correct? A ( cdot alpha=2 beta ) в. ( alpha=beta ) c. ( 2 alpha=beta ) |
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| 157 | The value of ( ^{n} C_{0}+3 times^{n} C_{1}+9 times^{n} ) ( boldsymbol{C}_{2}+ldots+boldsymbol{3}^{n} times^{n} boldsymbol{C}_{n} ) A ( cdot 2^{n} ) B. ( 3^{n} ) ( c cdot 4^{n} ) D. ( 5^{n} ) |
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| 158 | (1982 – 2 Marks) The sum of the coefficients of the plynomial (1+x -3×2 2163 is ……… (1982-2 Marlo IF(1 Iarn -119.242 |
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| 159 | The ( 3 r d, 4 t h, ) and 5 th terms in the expansion ( (x+a)^{n} ) are respectively ( 84,280, ) and ( 560, ) find the values of ( x, a ) and ( n ) A. ( x=1, a=2, n=6 ) B. ( x=1, a=6, n=7 ) c. ( x=3, a=2, n=7 ) D. ( x=1, a=2, n=7 ) |
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| 160 | Find the term independent of ( ^{prime} x^{prime} ) in the expansion of the expression, ( (1+x+ ) ( left.2 x^{3}right)left(frac{3}{2} x^{2}-frac{1}{3 x}right)^{9} ) |
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| 161 | ff ( f(x)=x^{4}+10 x^{3}+39 x^{2}+76 x+65 ) find the value of ( f(x-4) ) |
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| 162 | Find the coeffcient of: ( x^{-7} ) in the expansion of ( left(a x-frac{1}{b x^{2}}right)^{8} ) ii) ( x^{6} ) in the expansion ( left(a-b x^{2}right)^{10} ) |
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| 163 | If the fourth term in the expansion of ( (p x+1 / x)^{n} ) is ( 5 / 2 ) then the value of ( p ) is ( A ) B. 1/2 ( c cdot 6 ) D. |
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| 164 | The term independent of ( x ) in the expansion of ( left(1+x+2 x^{3}right)left(frac{3 x^{2}}{2}-right. ) ( left.frac{1}{3 x}right)^{9} ) is A ( cdot frac{13}{54} ) в. ( frac{15}{54} ) c. ( frac{17}{54} ) D. ( frac{19}{54} ) |
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| 165 | The coefficient of ( x^{5} ) in the expansion of ( left(1+x^{2}right)^{5}(1+x)^{4} ) is ( ? ) ( mathbf{A} cdot 61 ) B. 59 c. zero D. 60 |
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| 166 | The middle term of expansion of ( left(frac{10}{x}+frac{x}{10}right)^{10} ) A ( cdot^{7} C_{5} ) в. ( ^{8} C_{5} ) ( mathrm{c} cdot^{9} mathrm{C}_{5} ) D. ( ^{10} C_{5} ) |
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| 167 | Find the term of expansion of ( left(x+frac{1}{x}right)^{n} ) which does not contain ( x ) | 11 |
| 168 | Let ( n ) be a positive integer. If the coefficients of ( 2^{n d}, 3^{r d} ) and ( 4^{t h} ) terms in the expansion of ( (1+x)^{n} ) are in A.P. then the value of ( n ) is: ( mathbf{A} cdot mathbf{8} ) B. 27 c. 12 D. |
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| 169 | Write the middle terms in the expansion of ( left(frac{3 x}{7}-2 yright)^{10} ) |
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| 170 | ( (sqrt{3}+sqrt{2})^{4}-(sqrt{3}-sqrt{2})^{4}= ) A ( .20 sqrt{6} ) в. ( 30 sqrt{6} ) c. ( 5 sqrt{10} ) D. ( 40 sqrt{6} ) E ( .10 sqrt{6} ) |
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| 171 | The greatest value of the term independent for ( x ) in the expansion of ( left(x sin p+x^{-1} cos pright)^{10}, p in R, ) is ( A cdot 2^{5} ) в. ( frac{10 !}{(5 !)^{2}} ) c. ( frac{1}{2^{5}} cdot frac{10 !}{(5 !)^{2}} ) D. None of the above |
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| 172 | distinct primes, then show that In nk ln2 (1984-2 Marks) Find the sum of the series : 1 31 7 C – +- +- +- Ar ….. up to m terms] 27 22r Č (–19 “C,13+ 2 + 2 + y pu.. up to m terms] 237 x 15 24r ….. Un |
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| 173 | | (4) O2 (U) – 1 23. If n is a positive integer , then (13+1)?” -(13 – 1)?” is: [2012] (a) an irrational number (6) an odd positive integer an even positive integer (d) a rational number other than positive integers (C) an even |
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| 174 | Number of terms free from radical sign in the expansion of ( left(1+3^{1 / 3}+7^{1 / 2}right)^{10} ) is ( A cdot 4 ) B. 5 ( c cdot 6 ) D. |
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| 175 | Find the value(s) of k such that the term independent of ( x ) in ( left(3 x^{2}+frac{k}{2 x}right)^{6} ) is 135 ( A cdot pm 2 ) B. ±1 ( c .pm 3 ) D. ±4 |
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| 176 | Let ( [x] ) denote the greatest integer part of a real number x. If ( boldsymbol{M}=sum_{n=1}^{40}left[frac{boldsymbol{n}^{2}}{mathbf{2}}right] ) then m equals A . 5700 B. 5720 ( c .5740 ) D. 11060 |
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| 177 | Sum of coefficients in the expeansion of ( (a+b+c)^{8} ) is A. 2154 в. 6561 c. 729 D. 1944 |
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| 178 | The sum of the coefficients of first three terms in the expansion of ( left(x-frac{3}{x^{2}}right)^{m}, x neq 0, m ) being a natura number, is ( 559 . ) Find the term of the expansion containing ( boldsymbol{x}^{mathbf{3}} ) |
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| 179 | Find the negative of middle term in the expansion of ( left(frac{2 x}{3}-frac{3}{2 x}right)^{6} ) |
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| 180 | The first 3 terms in the expansion of ( (1+a x)^{n}(n neq 0) ) are ( 1,6 x ) and ( 16 x^{2} ) Then the value of ( a ) and ( n ) are respectively A . 2 and 9 B. 3 and 2 c. ( 2 / 3 ) and 9 D. ( 3 / 2 ) and 6 |
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| 181 | If ( C_{0} . C_{1}, C_{2}, dots, C_{n} ) are the coefficients of the expansion of ( (1+x)^{n}, ) then the value of ( sum_{0}^{n} frac{C_{k}}{k+1} ) is A . в. ( frac{2^{n}-1}{n} ) c. ( frac{2^{n+1}-1}{n+1} ) D. None of these |
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| 182 | ( operatorname{Let}left(1+x^{2}right)^{2}(1+x)^{n}=A_{0}+A_{1} x+ ) ( A_{2} x^{2}+ldots . ) If ( A_{0}, A_{1}, A_{2} ) are in A.P, then the value of ( n ) ( A cdot 2 ) B. 3 ( c cdot 5 ) D. 7 |
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| 183 | The middle term in the expansion of ( left(frac{x}{y}+frac{y}{x}right)^{8} ) is ( A cdot^{8} C_{5} ) в. ( ^{8} mathrm{C}_{6} ) ( mathrm{c} cdot^{8} mathrm{C}_{4} ) D. ( ^{8} mathrm{C}_{2} ) |
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| 184 | 2 4 27. If the number of terms in the expansion of 1-+- X+0, is 28, then the sum of the coefficients of all the term in this expansion, is : JJEEM 2016 (a) 243 (b) 729 (c) 64 (d) 2187 |
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| 185 | Find the middle term in the expansion of : ( left(frac{x}{a}-frac{a}{x}right)^{10} ) |
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| 186 | Find the greater number in ( 300 ! ) and ( sqrt{300^{300}} ) | 11 |
| 187 | Find the number of terms in the expansion of ( left(1-2 x+x^{2}right)^{7} ) |
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| 188 | The term independent of ( x ) in the expansion of ( left(x^{2}-frac{3 sqrt{3}}{x^{3}}right)^{10} ) | 11 |
| 189 | Using binomial theorem find the value of ( (102)^{3} ) | 11 |
| 190 | Find the term independent of ( x ) in ( (x+ ) ( left.frac{1}{x}right)^{4} ) | 11 |
| 191 | 14. If x is so small that rand higher powers of x may be (1 + x)2 – 1+ neglected, then may be approximated as (1-x)2 (a) 1-3×2 (b) 3x + 2×2 [2005] 8 12 to 8 |
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| 192 | 6. If x is positive, the first negative term in the expansion of (1+x) 27/5 is [2003] (a) 6th term (b) 7th term (c) 5th term (d) 8th term. |
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| 193 | The value of the expansion ( (sqrt{3}+1)^{5}+ ) ( (sqrt{3}-1)^{5} ) ( mathbf{A} cdot 88 ) B . 40 c. ( 88 sqrt{3} ) D. ( 40 sqrt{3} ) |
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| 194 | Find the second term of the binomial expansion of ( left(sqrt[13]{boldsymbol{a}}+frac{boldsymbol{a}}{sqrt{boldsymbol{a}^{-1}}}right)^{m}, ) if ( boldsymbol{C}_{3}^{boldsymbol{m}} ) ( C_{2}^{m}=4: 1 ) |
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| 195 | The third term from the end in the expansion of ( left(frac{4 x}{3 y}-frac{3 y}{2 x}right)^{9} ) is A ( cdot ) s ( _{C_{7}} frac{3^{5}}{2} frac{y^{5}}{x^{5}} ) В. ( -_{-9} sigma_{7} frac{3^{5}}{2^{3}} frac{y^{5}}{x^{5}} ) c. ( _{9} C_{7} frac{3^{5}}{2^{3}} frac{y^{5}}{x^{3}} ) D. none of these |
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| 196 | Using Binomial Theorem, evaluate ( (101)^{4} ) | 11 |
| 197 | ( left(begin{array}{l}n \ 0end{array}right)+2left(begin{array}{l}n \ 1end{array}right)+2^{2}left(begin{array}{l}n \ 2end{array}right)+ldots .+2^{n}left(begin{array}{l}n \ nend{array}right) ) is equal to A ( cdot 2^{n} ) B. ( c cdot 3^{n} ) D. None of these |
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| 198 | If the coefficients of rth, (r+1)th, and (r+2)th terms in the the binomial expansion of (1+y)” are in A.P., then mand r satisfy the equation [2005] (a) m? – m(4r-1)+4r2 – 2 = 0 (b) m2 – m (4r+1)+4 r2 +2=0 (c) m2 – m (4r+1)+ 4 r2 – 2=0 (d) m? –m (4r-1)+4 2 +2 = 0 |
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| 199 | If in the expansion of ( (1+x)^{43}, ) the coefficient of ( (2 r+1) t h ) term is equal to coefficient of ( (r+2)^{t h} ) term. Find ( r ) ?? |
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| 200 | ( operatorname{Let}left(frac{2 x^{2}+x+2}{x}right)^{n}=sum_{r=m}^{r=t} a_{r} x^{r} ) then answer the following: ( fleft(a_{p}=a_{q} text { then } p+q=dotsright. ) |
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| 201 | The coefficient of ( x^{5} ) in the expansion of ( left(x^{2}-x-2right)^{5} ) is A. 351 в. -82 c. -86 D. -81 |
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| 202 | If ( boldsymbol{X}=left{4^{n}-3 n-1: n in Nright} ) and ( boldsymbol{Y}={mathbf{9}(boldsymbol{n}-mathbf{1}): boldsymbol{n} in boldsymbol{N}}, ) where ( mathrm{N} ) is the set of natural numbers, then ( boldsymbol{X} cup boldsymbol{Y} ) is equal to: A. ( Y ) B. ( N ) c. ( Y-x ) D. ( x ) |
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| 203 | Write down and simplify: The 25 th term of ( (5 x+8 y)^{30} ) |
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| 204 | The coefficient of ( x ) in the expansion of ( (1-a x)^{-1}(1-b x)^{-1}(1-c x)^{-1} ) is? A ( . a+b+c ) B. ( a-b-c ) c. ( -a+b+c ) D. ( a-b+c ) |
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| 205 | Find the middle term in the expansion of ( left(frac{x}{a}-frac{a}{x}right)^{21} ) A ( cdot 20_{110} frac{x}{a},^{21} C_{10} frac{a}{x} ) В. ( _{20} C_{9} frac{x}{a},^{2} 16_{10} frac{a}{x} ) ( ^{mathrm{C}} cdot_{21} C_{10} frac{x}{a^{2}},-^{21} 1_{10} frac{a}{x} ) D. ( _{21} C_{9} frac{x}{a},^{21} C_{10} frac{a}{x} ) |
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| 206 | f ( x+y=1, ) then ( sum_{r=0}^{n} r^{n} C_{r} x^{r} . y^{n-r}= ) ( mathbf{A} cdot mathbf{1} ) B. c. ( n x ) D. ( n y ) |
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| 207 | The total number of terms in the expansion of ( (x+y)^{50}+(x-y)^{50} ) is A . 51 B . 26 ( c .102 ) D. 25 |
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| 208 | Show that the middle term in the expansion of ( (1+x)^{2 n} ) is ( frac{1.3 .5 ldots . . .(2 n-1)}{n !} ) ( 2^{n} x^{n} ; ) where ( n ) is a positive integer. |
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| 209 | The third term from the end in the expansion of ( left(frac{3 x}{5}-frac{5}{2 x}right)^{8} ) is A ( cdot frac{35451}{15 x^{4}} ) в. ( frac{45455}{16 x^{4}} ) c. ( frac{39372}{15 x^{4}} ) D. ( frac{39375}{16 x^{4}} ) |
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| 210 | Expand the following binomial ( left(1-frac{1}{x}right)^{10} ) | 11 |
| 211 | ( operatorname{If}left(1+2 x+x^{2}right)^{n}=sum_{r=0}^{2 n} a_{r} x^{r}, ) then ( a_{r}= ) A ( cdotleft(^{n} C_{r}right)^{2} ) В. ( ^{n} C_{r} cdot^{n} C_{r+1} ) ( c cdot^{2 n} C_{r} ) D. ( ^{2 n} C_{r+1} ) |
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| 212 | Number of irrational terms in the expansion of ( (sqrt{2}+sqrt{3})^{15} ) is equal to A . 16 B. 7 c. 12 D. 15 |
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| 213 | The expansion of ( left[boldsymbol{x}+left(boldsymbol{x}^{mathbf{3}}-mathbf{1}right)^{mathbf{1} / 2}right]^{mathbf{5}}+ ) ( left[x-left(x^{3}-1right)^{1 / 2}right]^{5} ) is a polynomial of degree ( mathbf{A} cdot mathbf{8} ) B. 7 ( c cdot 6 ) D. 5 |
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| 214 | In the expansion of ( left(3^{-x / 4}+3^{5 x / 4}right)^{n} ) the sum of binomial coefficient is 64 and term with the greatest bionomial coefficient term exceeds the third term by ( (n-1) ) the value of ( x ) must be ( A cdot 0 ) B. ( c cdot 2 ) D. 3 |
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| 215 | Assertion No three consecutive binomial coefficient can be in G.P. & H.P. Reason Three consecutive binomial coefficients are in A.P. A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion, B. Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion, c. Assertion is true but Reason is false D. Assertion is false but Reason is true |
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| 216 | Suppose that ( n ) is a natural number and ( boldsymbol{I}, boldsymbol{F} ) are respectively the integral part and fractional part of ( (7+4 sqrt{3})^{n} . ) Then show that i) ( I ) is an odd integer ii) ( (boldsymbol{I}+boldsymbol{F})(mathbf{1}-boldsymbol{F})=mathbf{1} ) |
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| 217 | If the 7 th and 8 th term of the binominal expansion ( (2 a-3 b)^{n} ) are equal, then ( frac{2 a+3 b}{2 a-3 b} ) is to A ( cdot frac{n-13}{n+1} ) в. ( frac{n+1}{13-n} ) c. ( frac{6-n}{13-n} ) D. ( frac{n-1}{13-n} ) |
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| 218 | In the binomial expansion of ( (1+y)^{n} ) where ( n ) is a natural number, the coefficients of the ( 5^{t h}, 6^{t h} ) and ( 7^{t h} ) terms are in A.P, find ( n ) This question has multiple correct options ( mathbf{A} cdot n=7 ) B . ( n=14 ) c. ( n=8 ) D. ( n=16 ) |
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| 219 | The value of ( left(10^{11}-10^{9}-2 times 11 times 10^{8}-3 times 11^{2} timesright. ) is equal to ( mathbf{A} cdot mathbf{0} ) B. ( 11^{10} ) ( c cdot 11^{11} ) D. ( 10^{11} ) |
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| 220 | In the expansion of ( left(x-frac{1}{x}right)^{6} ), the constant term is A . -20 B . 20 c. 30 D. -30 |
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| 221 | The coefficient of the term independent of ( x ) in the expansion of ( left(frac{x+1}{x^{frac{2}{3}}-x^{frac{1}{3}}+1}-frac{x-1}{x-x^{frac{1}{2}}}right)^{10} ) A . 210 в. 105 c. 70 D. 112 |
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| 222 | If the fourth term in the expansion of ( left(sqrt{frac{1}{x log x+1}}+x frac{1}{12}right)^{6} ) is equal to 200 and ( x>1, ) then ( x ) is A . 10 B. ( 10^{-4} ) ( c cdot 1 ) D. -4 |
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| 223 | The co-efficient of ( x^{53} ) in the expression ( sum_{m=0}^{100} 100 c_{m}(x-3)^{100-m} 2^{m} ) is B. ( 98_{c_{53}} ) ( mathbf{c} .^{65} sigma_{53} ) D. ( 100 c_{65} ) |
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| 224 | The coefficient of ( t^{24} ) in the expansion of ( left(1+t^{2}right)^{12}left(1+t^{12}right)left(1+t^{24}right) ) is A. ( ^{12} C_{6}+2 ) B. ( ^{12} C_{5} ) ( mathbf{c} cdot^{12} C_{6} ) D. ( ^{12} C_{7} ) |
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| 225 | Find the ( (p+2) ) th term from the end in ( left(x-frac{1}{x}right)^{2 n+1} ) | 11 |
| 226 | The value of ( x ) in the expression ( left(x+x^{log _{10} x}right)^{5}, ) if the third term in the expansion is ( 10,00,000, ) is This question has multiple correct options A ( cdot 10^{-1} ) B . ( 10^{text {। }} ) ( mathbf{c} cdot 10^{-5 / 2} ) D. ( 10^{5 / 2} ) |
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| 227 | For positive integers ( n_{1}, n_{2} ) the value of the expression ( (1+i)^{n_{1}}+left(1+i^{3}right)^{n_{1}}+ ) ( left(1+i^{5}right)^{n_{2}}+left(1+i^{7}right)^{n_{2}}, i=sqrt{-1} ) is a real number if and only if ( mathbf{A} cdot n_{1}=n_{2}+1 ) В . ( n_{1}=n_{2}-1 ) c. ( n_{1}=n_{2} ) D ( . n_{1}>0, n_{2}>0 ) |
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| 228 | If ( boldsymbol{X}=left{mathbf{8}^{n}-mathbf{7} boldsymbol{n}-mathbf{1}, boldsymbol{n} in boldsymbol{N}right} ) and ( boldsymbol{Y}= ) ( mathbf{4} 9(n-1), n in N, ) then ( (operatorname{given} n>1) ) A. ( X subset Y ) в. ( Y subset X ) c. ( X=Y ) D. ( X nsubseteq Y ) |
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| 229 | The positive integerjust greater than ( (1+0.0001)^{10000} ) is ( A cdot 4 ) B. 5 ( c cdot 2 ) D. |
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| 230 | The sum of the coefficients in the first, second, and third terms of the expansion of ( left(x^{2}+frac{1}{x}right)^{m} ) is equal to 46 Find the term of the expansion which does not contain ( x ) |
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| 231 | If ( (1+a x)^{n}=1+8 x+24 x^{2}+dots ) then ( boldsymbol{a} times boldsymbol{n} ) is: ( mathbf{A} cdot mathbf{8} ) B. 12 c. 16 D . 24 |
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| 232 | Evaluate ( (sqrt{3}+sqrt{2})^{6}-(sqrt{3}-sqrt{2})^{6} ) | 11 |
| 233 | 19. Let n be a positive integer and (1994 – 5 (1 + x + x2)n = a+a, x+ …………+ a), x2 Show that a 2-a,2+ a,2 ………….+ a,,2=an |
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| 234 | Find the expansion ( left(3 x^{2}-2 a x+3 a^{2}right)^{3} ) using binomial theorem. |
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| 235 | The value of the term independent of ( x ) in the expansion of ( left(x^{2}-frac{1}{x}right)^{27} ) is : ( mathbf{A} cdot mathbf{9} ) B. 18 c. 48 D. 84 |
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| 236 | The number of integral terms in ( (sqrt{3}+sqrt[8]{2})^{64} ) is ( mathbf{A} cdot mathbf{8} ) B. 7 ( c .9 ) D. 6 |
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| 237 | If for ( 1 leq m leq n, f(m, n)=C_{0}-C_{1}+ ) ( C_{2}-ldots .(-1)^{m-1} C_{m-1}, ) find ( f(9,5) ) |
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| 238 | ( 5^{t h} ) term from the end in the expansion of ( left(frac{x^{2}}{2}-frac{2}{x^{2}}right)^{12} ) is B . ( 7920 x^{4} ) c. ( 7920 x^{-4} ) D. ( -7920 x^{4} ) |
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| 239 | The term independent of ( x ) in the expansion of ( left(x-frac{1}{x}right)^{4}left(x+frac{1}{x}right)^{3} ) is A . -3 B. ( c .3 ) D. |
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| 240 | Find the fourteenth term of ( (3-a)^{15} ) | 11 |
| 241 | If the coefficient of three consecutive terms in the expansion of ( (1+x)^{n} ) be ( 165,330, ) and ( 462 . ) Find ( n ) |
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| 242 | ( (1-sqrt{2})^{6}= ) A ( .98-70 sqrt{2} ) В. ( 99-70 sqrt{2} ) D. ( 98+70 sqrt{2} ) |
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| 243 | 9. The coefficient of the middle term in the binomial expansion in powers of x of (1+ ax)4 and of (1 – ax) is the same if a equals [2004] (a) 5 (6) (b) (c) To (d) – |
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| 244 | The coefficient of ( x^{9} ) in the expansion of ( left(x^{3}+frac{1}{2^{t}}right)^{11}, ) where ( t=log _{sqrt{2}}left(x^{frac{3}{2}}right) ) A . -5 в. 330 ( c .520 ) D. ( 5+log _{sqrt{2}}(3) ) |
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| 245 | Find the term independent of ( x ) in the expansion of ( left(sqrt{frac{boldsymbol{x}}{mathbf{3}}}+frac{mathbf{3}}{mathbf{2} boldsymbol{x}^{2}}right)^{10} ) ( mathbf{A} cdot T_{3} ) в. ( T_{4} ) c. ( T_{5} ) D. No term will be independent of ( x ) |
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| 246 | In the expansion of ( left(x^{3}-frac{1}{x^{2}}right)^{2}, ) where n is a positive integer, the sum of the coefficients of ( boldsymbol{x}^{boldsymbol{6}} ) is ( mathbf{1} ) |
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| 247 | The 2 nd, 3 rd and 4 th terms in the expansion of ( (x+y)^{n} ) are 240,720,1080 respectively; find ( x, y, n ) |
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| 248 | Solve ( :left(frac{2 n}{2 n-1}right)^{p}=left(frac{1}{1-left(frac{p}{2 n}right)}right)^{p} ) | 11 |
| 249 | ( f^{n-1} C_{r}=left(k^{2}-3right)^{n} C_{r+1}, ) then ( k ) B. ( [2, infty) ) c. ( [-sqrt{3}, sqrt{3}] ) D. ( (sqrt{3}, 2] ) |
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| 250 | In the expansion of ( left(x^{2}-frac{1}{4}right)^{n}, ) the coefficient of third term is ( 31, ) then the value of ( n ) is- A . 30 B. 31 c. 29 D. 32 |
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| 251 | ff ( left(x^{2}+frac{1}{x}right)^{n} ) has exactly one middle term which is equal to ( alpha . x^{3} ) then the value of ( (boldsymbol{alpha}+boldsymbol{n}) ) is- ( quad(boldsymbol{n} in boldsymbol{N}) ) A . 18 B . 21 c. 24 D. 26 |
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| 252 | 3. The positive integer just greater than (1 +0.0001)10000 is [2002] (2) 4 (6) 5 (c) 2 (d) 3 |
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| 253 | Expand ( left(2 x^{2}+3right)^{4} ) |
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| 254 | If ( (5+2 sqrt{6})^{n}=m+f ), where ( n ) and ( m ) are positive integers and ( 0 leq f<1 ) then ( frac{1}{1-f}-f ) is equal to A ( cdot frac{1}{m} ) в. ( m ) c. ( _{m+frac{1}{m}} ) D. ( _{m-} frac{1}{m} ) |
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| 255 | If the third term in the expansion of ( left[x+x^{log _{10} x}right]^{5} ) is ( 10^{6}, ) then ( x ) can be This question has multiple correct options A ( cdot 10^{-1 / 3} ) B. 10 c. ( 10^{-5 / 2} ) D. ( 10^{2} ) |
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| 256 | The middle term in the expansion of ( left(frac{2 x}{3}-frac{3}{2 x^{2}}right)^{2 n} ) is A ( cdot 2^{n} mathrm{C}_{n} ) B cdot ( (-1)^{n}left[(2 n !) /(n !)^{2}right] cdot x^{-n} ) ( mathrm{c} cdot_{2 n} mathrm{C}_{n} cdot frac{1}{x^{n}} ) D. none of these |
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| 257 | Expansion of ( (boldsymbol{y}+boldsymbol{x})^{n} ) is | 11 |
| 258 | The number of rational terms in the expansion of ( left(x^{frac{1}{5}}+y^{frac{1}{10}}right)^{45} ) is A. 5 B. 6 ( c cdot 4 ) D. |
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| 259 | If ( T_{r} ) denotes the ( r^{t h} ) term in the expansion of ( left(x+frac{1}{x}right)^{23}, ) then ( mathbf{A} cdot T_{12}=T_{13} ) В . ( x^{2} . T_{13}=T_{12} ) c. ( x^{2} . T_{12}=T_{13} ) D. ( T_{12}+T_{13}=25 ) |
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| 260 | e sum of the co-efficients of all odd degree terms in the expansion of (x+Vx3 -1)3 +(x-Vx3-1)”,(x>1) is : [JEEM 2018] (a) o (6) 1 do (c) 2 (d) – 1 |
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| 261 | The middle term in the expansion of ( left(x+frac{1}{x}right)^{10} ) A ( cdot ) io ( _{1} frac{1}{x} ) в. ( ^{10} C_{5} ) c. ( ^{10} C_{6} ) D. ( ^{10} C_{7} x ) |
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| 262 | Find the middle term in the expansion of ( left(frac{2 x}{3}+frac{3}{2 x}right)^{10} ) A . 210 в. 630 ( c .252 ) D. 756 |
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| 263 | UL. TUU lay uuu 23. Let n be any positive integer. Prove that (1999 – 10 Marks) 2n-k mlk n-k) 2n-2m (2n-4k+1) 9n-2k = on-Ak (2n-2k +1) (2n – 2m) In-m kao for each non-be gatuve integer msn. | Here |
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| 264 | Find the sixth term in the expansion of ( left(2 x^{2}-frac{3}{7 x^{3}}right)^{11} ) A ( cdot-^{11} C_{5} frac{2^{6} 3^{5}}{7^{5}} x^{3} ) В ( cdot quad^{11} C_{5} frac{2^{6} 3^{5}}{7^{5}} x^{-3} ) c. ( _{-11} C_{5} frac{2^{6} 3^{5}}{7^{5}} x^{-3} ) D. None of these |
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| 265 | Find coefficient of ( a^{3} b^{4} c^{5} ) in the expansion of ( (b c+c a+a b)^{6} ) |
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| 266 | In the expansion of ( left(7^{1 / 3}+11^{1 / 9}right)^{6561} ) prove that three will be only 730 term which are free from radicals |
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| 267 | In the expansion of ( (1+x)^{n}, ) the binomial coefficients of 3 consecutive terms are respectively 220,495 and 792 then ( n= ) A .4 B. 8 ( c cdot 12 ) D. 16 |
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| 268 | If the constant term of the binomial expansion ( left(2 x-frac{1}{x}right)^{n} ) is ( -160, ) then ( n ) is equal to – A .4 B. 6 c. 8 D. 10 |
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| 269 | Expand to 4 terms the following expressions: ( (1+x)^{frac{2}{5}} ) |
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| 270 | f ( log 1001=3.000434 ), find the number of digits in ( 1001^{101} ) | 11 |
| 271 | Expansion of ( (3 x+2)^{3} ) is ( 27 x^{3}+8+ ) ( 18 x(3 x+2) ) A. True B. False |
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| 272 | The tenth term in the expansion of ( left(2 x^{2}+frac{1}{x}right)^{12} ) A ( cdot frac{1760}{x^{3}} ) в. ( -frac{1760}{x^{3}} ) c. ( frac{1760}{x^{2}} ) D. none of the above |
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| 273 | 3 17. 145, 35) In the binomial expansion of (a – b)”, n > 5, the sum of [2007] and 6th terms is zero, then a/b equals n-5 a n-4 o 5 (d) (a)” (6) “5* ©n-4 n-5 20071 |
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| 274 | Write general terms of this ( 2 xleft[3+2 x^{2}right]^{20} ) |
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| 275 | The coefficient of ( x^{11} ) in the expansion of ( left(1-2 x+3 x^{2}right)(1+x)^{11} ) is A . 164 в. 144 c. 116 D. none of these |
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| 276 | The coeficient of x’in ( The coefficient of x4 in is (1983 – 1 Mark) “(19831 Mar (b) 504 504 (a) 405 256 259 450 263 (d) none of these |
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| 277 | 1200T 18. The sum of the series 20 Co – 20G + 2002 – 2003 +… -.+ 206, is (a) o (6) 20. c) _2000 (a) 220610 |
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| 278 | Expand ( left(boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}right)^{boldsymbol{6}} ) | 11 |
| 279 | Find the middle term(s) in the expansion of ( left(1+3 x+3 x^{2}+x^{3}right)^{2 n} ) |
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| 280 | If ( left(1+x^{2}right)^{2}(1+x)^{n}=C_{0}+C_{1} x+ ) ( C_{2} x^{2}+cdots, ) and if ( C_{0}, C_{1}, C_{2} ) are in A.P. find ( n ) This question has multiple correct options A .2 B. 3 ( c cdot 4 ) D. 5 |
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| 281 | The middle term in the expansion of ( left(x+frac{1}{x}right)^{10} ) A ( cdot ) io ( C_{4} cdot frac{1}{x} ) в. ( ^{10} C_{5} ) c. ( _{10} mathrm{C}_{5} . frac{1}{x} ) D. ( ^{10} C_{6} . x ) |
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| 282 | The greatest value of the term independent of ( x, ) as ( alpha ) varies over ( R ) in the expansion of ( left(x cos alpha+frac{sin alpha}{x}right) ) is B. ( ^{20} C_{19} ) c. ( 20 C_{6} ) D. ( ^{20} C_{6}left(frac{1}{2}right)^{10} ) |
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| 283 | If the sum of the coefficients in the expansion of (a + b)” is 4096, then the greatest coeficient in the expansion is [2002] (a) 1594 (b) 792 (c) 924 (d) 2924 anni 10000. |
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| 284 | Find the middle term in the expansion of ( left(1-2 x+x^{2}right)^{n} ) A ( cdot frac{(2 n) !}{(n !)^{2}}(-1)^{n} x^{n} ) B. ( frac{(2 n) !}{(n !)}(-1)^{n} x^{2} n ) c. ( frac{(2 n) !}{(n !)^{2}} x^{n} ) D. None of these |
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| 285 | Find the coefficients of ( x^{2} ) and ( x^{3} ) in the expansion of ( (2-x)^{6} ) |
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| 286 | Write the general term in the expansion of ( left(x^{2}-yright)^{6} ) |
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| 287 | The ratio of coefficient of ( x^{3} ) and ( x^{4} ) in expansion ( (1+x)^{12} ) is: ( A cdot frac{4}{9} ) B . ( frac{1}{3} ) ( c cdot frac{2}{3} ) D. |
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| 288 | και η 5. Σε αυτά της – Σ ε εθει με τον ερχολ το ΥΞΟ 2n -1 (2004) (c) n – 1 |
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| 289 | Which of the following binomial expressions has a least term independent of ( x ? ) ( ^{mathrm{A}} cdotleft(sqrt{x}-frac{3}{x^{2}}right)^{1} ) B. ( left(x+frac{1}{x}right)^{6} ) c. ( (1+x)^{32} ) ( ^{mathrm{D}}left(frac{3}{2} x^{2}-frac{1}{3 x}right)^{9} ) |
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| 290 | Find the coefficient of ( x^{17} ) in ( (x+y)^{20} ? ) | 11 |
| 291 | If the middle term in the expansion of ( left(x^{2}+frac{1}{x}right)^{n} ) is ( 924 x^{6}, ) then ( n= ) A . 10 B. 12 c. 14 D. none of these |
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| 292 | The term independent of ( x ) in the expansion of ( left(boldsymbol{x}^{2}-frac{mathbf{1}}{boldsymbol{x}}right)^{boldsymbol{6}} ) is A . -12 B. 15 ( c cdot 24 ) D. -15 |
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| 293 | The coefficient of ( t^{50} ) in ( left(1+t^{2}right)^{25}left(1+t^{25}right)left(1+t^{40}right)left(1+t^{45}right)(1 ) is A ( cdot 1+sqrt[25]{5} ) B. ( 1+^{25} C_{5}+^{25} C_{7} ) ( mathbf{C} cdot 1+^{25} C_{7} ) D. None of these |
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| 294 | What is the unit digit in the product ( left(3^{65} times 6^{59} times 7^{71}right) ? ) ( mathbf{A} cdot mathbf{1} ) B . 2 ( c cdot 4 ) D. 6 |
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| 295 | The middle term in the expansion of ( left(1-3 x+3 x^{2}-x^{3}right)^{2 n} ) is ( mathbf{A} cdot 6 n_{C_{3 n}}(-x)^{3 n} ) B . ( 6 n_{C_{2 text { n }}}(-x)^{2 n+1} ) ( mathbf{c} cdot 4 n_{C_{S n}}(-x)^{3 n} ) D. ( 6 n_{C_{3 n}}(-x)^{3 n-1} ) |
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| 296 | The middle term in the expansion of ( (1+x)^{2 n} ) is A. ( frac{1.3 .5 ldots(2 n-1)}{n} x^{n} ) B. ( frac{1.3 .5 ldots(2 n-1)}{n !} 2^{n-1} x^{n} ) c. ( frac{1.3 .5 ldots(2 n-1)}{n !} x^{n} ) D. ( frac{1.3 .5 ldots(2 n-1)}{n !} 2^{n} x^{n} ) |
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| 297 | Find the middle term in the expansion of ( left(frac{2}{3} x-frac{3}{2 x}right)^{20} ) A ( .^{20} C_{10} x^{10} y^{10} ) B. ( ^{20} C_{11} x^{11} y^{11} ) C. ( ^{20} C_{9} x^{11} y^{10} ) D. None of these |
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| 298 | f ( ^{n} C_{4},^{n} C_{5},^{n} C_{6} ) of ( (1+x)^{n} ) are in A.P. then ( n= ) A . 12 B. 1 ( c cdot 7 ) ( D .8 ) |
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| 299 | Expand the expression ( (2 x-3)^{6} ) | 11 |
| 300 | Coefficient of ( boldsymbol{x}^{boldsymbol{9}} ) in ( rightarrow(mathbf{1}+boldsymbol{x})(mathbf{( 1}+boldsymbol{t}) ) ( left.left.boldsymbol{x}^{2}right)left(mathbf{1}+boldsymbol{x}^{3}right) ldots ldotsleft(boldsymbol{1}+boldsymbol{x}^{mathbf{1 0 0}}right)right) ? ) |
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| 301 | 14. F E 4, (x-2)* = 6, (x – 3)” and ax = 1 for all p=0 (1992 – 6 Marks) k > n, then show that b, = 2NFC |
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| 302 | 19. Statement-1: “C, = (n + 2)2n-1 [2008] r=0 Statement-2: E(r+1) “Cyx” = (1 + x)” + nx(1+x)”-. r=0 (a) Statement-1 is false, Statement-2 is true (b) Statement-1 is true, Statement-2 is true; Statement -2 15 a correct explanation for Statement-1 Statement -1 is true, Statement-2 is true; Statement -2 is not a correct explanation for Statement-1 (d) Statement -1 is true, Statement-2 is false |
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| 303 | Find the term of the expansion of ( (sqrt[3]{x^{-2}}+x)^{7} ) containing ( x ) in the second power ( mathbf{A} cdot T_{4} ) в. ( T_{5} ) c. ( T_{6} ) ( mathbf{D} cdot T_{7} ) |
11 |
| 304 | The number of non-zero terms in the expansion of ( (sqrt{7}+1)^{75}-(sqrt{7}-1)^{75} ) is A . 36 B. 37 c. 38 D. 39 |
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| 305 | ( 9^{t h} ) term in the expansion of ( left(frac{x}{a}-frac{3 a}{x^{2}}right)^{12} ) A ( cdot^{12} C_{9} cdot 3^{9} x^{-12} a^{6} ) В. ( ^{12} C_{6} cdot 3^{8} x^{-16} a^{6} ) C ( .^{12} C_{4} cdot 3^{8} x^{-12} a^{4} ) D. none of the above |
11 |
| 306 | 24. The term independent of x in expansion of J JEEM 2013 x+1_ -_*-1 is (r2/3 – X1/3+1 x – x12 (a) 4 (b) 120 (C) 210 (d) 310 |
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| 307 | The middle term in the expansion of ( left(frac{10}{x}+frac{x}{10}right)^{10} ) A . ( ^{10} C_{5} ) в. ( ^{10} C_{6} ) c. ( _{10} mathrm{C}_{5} frac{1}{x^{10}} ) D. ( ^{10} C_{5} x^{10} ) E . ( ^{10} C_{5} 10^{10} ) |
11 |
| 308 | Find the sum of the coefficients of the terms of the expansion ( left(1+x+2 x^{2}right)^{6} ) |
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| 309 | Show that the middle term in the expansion of ( (1+x)^{2 n} ) is ( frac{1.3 .5 ldots(2 n-1)}{n !} 2^{n} x^{n}, ) where ( n ) is a positive integer. |
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| 310 | The value of ( (sqrt{5}+1)^{5}-(sqrt{5}-1)^{5} ) is: A . 252 в. 352 c. 452 D. 552 |
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| 311 | Prove that the coefficient of ( x^{n} ) in the expression of ( (1+x)^{2 n} ) is twice the coefficient of ( x^{n} ) in the expression of ( (1+x)^{2 n-1} ) |
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| 312 | The co-efficient of ( x^{5} ) in the expansion of ( (1+x)^{21}+(1+x)^{22}+dots dots+ ) ( (1+x)^{30} ) is: A . ( ^{51} C_{5} ) в. ( ^{9} C_{5} ) c. ( ^{31} C_{6}-^{21} C_{6} ) D. ( ^{30} C_{5}+^{20} C_{5} ) |
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| 313 | Find the value ( (sqrt{3}+1)^{4}+(sqrt{3}-1)^{4}=? ) | 11 |
| 314 | For ( boldsymbol{r}=mathbf{0}, mathbf{1}, mathbf{2},, dots mathbf{1 0} ) let ( boldsymbol{A}_{r}, boldsymbol{B}_{r} ) and ( boldsymbol{C}_{boldsymbol{r}} ) denote respectively the coefficient of ( boldsymbol{x}^{r} ) in the expansions of ( (1+x)^{10},(1+x)^{20} ) and ( (1+x)^{30} . ) Then ( sum_{r=1}^{10} A_{r}left(B_{10} B_{r}-C_{10} A_{r}right) ) is equal to A. ( B_{10}-C_{10} ) B . ( A_{10}left(B_{10}^{2}-C_{10} A_{10}right) ) c. D. ( C_{10}-B_{10} ) |
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| 315 | If the second term of the expansion ( left[boldsymbol{a}^{1 / 13}+frac{boldsymbol{a}}{sqrt{boldsymbol{a}^{-1}}}right]^{n} quad boldsymbol{i s} 14 boldsymbol{a}^{5 / 2}, ) then the value of ( frac{n_{mathbf{S}}}{n_{mathbf{C}_{2}}} ) is A .4 B. 3 c. 12 D. 6 |
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| 316 | In the expression of ( left(2^{x}+frac{1}{4^{x}}right)^{n} ) ratio of 2nd and third terms is given by ( t_{3} / t_{2}= ) 7 and the sum of the co-efficients of 2 nd and ( 3 mathrm{rd} ) term is ( 36, ) then the value of ( x ) is A ( frac{-1}{3} ) в. ( frac{-1}{2} ) ( c cdot frac{1}{3} ) D. |
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| 317 | If ( omega neq 1 ) is a cube root of unity and ( (omega+x)^{n}=1+12 omega+69 omega+ldots . ) then values of ( 4 n ) and ( omega ) respectively are A . 36,1 в. 12,2 c. ( 24,1 / 2 ) D. ( 18,1 / 3 ) |
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| 318 | In the expansion of ( left(7^{1 / 3}+11^{1 / 9}right)^{6561} ) prove that there will be only 730 terms which are free from radicals. | 11 |
| 319 | If the coefficients of ( x ) and ( x^{2} ) in the expansion of ( (1+x)^{m}(1-x)^{n} ) are 3 and -6 respectively. Find the values of ( m ) and ( n ) |
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| 320 | The coefficient of ( t^{4} ) in the expansion of ( left(1+t^{2}right)^{3} ) ( A ) B. 3 ( c cdot-3 ) D. – |
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| 321 | The highest term in the expansion of ( (2 sqrt{5}+sqrt[6]{7})^{6} ) is A. ( 800 sqrt{35} ) 55 в. ( 700 sqrt{35} ) c. ( 320 sqrt{5} ) D. ( 100 sqrt{7} ) |
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| 322 | If ( boldsymbol{f}(boldsymbol{x})=(mathbf{1}+boldsymbol{x})^{15}=boldsymbol{C}_{0}+boldsymbol{C}_{1} boldsymbol{x}+ ) ( C_{2} x^{2}+ldots+C_{15} x^{15}, ) then ( f(2)= ) ( mathbf{A} cdot 1^{15} ) B. ( 3^{15} ) ( c cdot 2^{15} ) D. None of these |
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| 323 | Consider the expansion ( left(x^{2}+frac{1}{x}right)^{15} ) Consider the following statements: 1. There are 15 terms in the given expansion. 2. The coefficient of ( x^{12} ) is equal to that of ( x^{3} ) Which of the statements is/are correct |
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| 324 | Find the ( 4^{t h} ) term in the expansion of ( (x-2 y)^{12} ) |
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| 325 | If ( c_{0}, c_{1}, c_{2}, ldots ldots c_{n} ) are the coefficients in the expansion of ( (1+x)^{n}, ) when ( n ) is a positive integer, prove that (1) ( c_{0}-c_{1}+c_{2}-c_{3}+dots dots+ ) ( (-1)^{r} c_{r}=(-1)^{r} frac{mid n-1}{|underline{r}| n-r-1} ) (2) ( c_{0}-2 c_{1}+3 c_{2}-4 c_{3}+dots dots+ ) ( (-1)^{n}(n+1) c_{n}=0 ) (3) ( c_{0}^{2}-c_{1}^{2}+c_{2}^{2}-c_{3}^{2}+dots dots+ ) ( (-1)^{n} c_{n}^{2}=0, ) or ( (-1)^{frac{n}{2}} c_{frac{n}{2}} ) according as ( n ) is odd or even. |
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| 326 | The ( 3 r d ) term of ( (2+sqrt{3})^{3} ) is A . 16 B. 17 c. 18 D. 19 |
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| 327 | (a) J (0) 1 8. Coefficient of 124 in (1 +12)12 (1+12) (1 + 24) is (20035) (a) 12Cg +3 (b) 12Cq+1 (c) 12C ‘(d) 12C7+2 (2004) 21. 1 |
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| 328 | Expand: ( left(frac{2}{x}-frac{x}{2}right)^{5} ) |
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| 329 | ( [mathrm{AS} 1] ) If ( boldsymbol{A}=frac{1}{3} boldsymbol{B} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{B}=frac{1}{2} boldsymbol{C}, ) then ( boldsymbol{A} ) ( B: C= ) A .1: 3: 6 B. 2:3:6 ( c cdot 3: 2: 6 ) D. 3: 1: 2 |
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| 330 | Using binomial theorem evaluate the following: ( (98)^{5} ) |
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| 331 | Find the middle terms in the expansion of ( left(2 x^{2}-frac{1}{x}right)^{7} ) В. ( -280 x^{5}, 560 x^{2} ) C ( .560 x^{5},-280 x^{2} ) D . ( 280 x^{5},-560 x^{2} ) |
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| 332 | prove that ( C_{0}^{2 n} C_{n}-^{2 n-2} C_{n}+ ) ( ^{2 n-4} C_{n}=2^{n} )where( C_{r}=^{n} C_{r} ) |
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| 333 | The coefficient of ( 1 / x ) in the expansion of ( (1+x)^{n}(1+1 / x)^{n} ) is A. ( frac{n !}{(n-1) !(n+1) !} ) B. ( frac{2 n !}{(n-1) !(n+1) !} ) c. ( frac{2 n !}{(2 n-1) !(2 n+1) !} ) D. none of these |
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| 334 | The middle term in the expansion of ( left(1-frac{1}{x}right)^{n}(1-x)^{n}, ) is A ( cdot^{2 n} C_{n} ) в. ( ^{-2 n} C_{n} ) ( mathrm{c} .^{-2 n} C_{n-1} ) D. none of these |
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| 335 | Using the formula for squaring a binomial the value of ( (999)^{2} ) is: A. 98009 B. 998005 c. 998001 D. 998002 |
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| 336 | Number of rational terms in the expansion of ( (sqrt{mathbf{2}}+sqrt[4]{mathbf{3}})^{100} ) is A . 25 B . 26 c. 27 D . 28 |
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| 337 | If in the expansion of ( (1+x)^{n}, ) the coefficients of three consecutive terms are ( 56,70,56, ) then the value of ( n ) and the position of the terms of these coefficients are given by A ( cdot n=8, ) the terms are ( 4^{t h}, 5^{t h}, 6^{t h} ) B . ( n=7 ), the terms are ( 3^{r d}, 4^{t h}, 5^{t h} ) C ( cdot n=8 ), the terms are ( 5^{t h}, 6^{t h}, 7^{t h} ) D. ( n=7 ), the terms are ( 4^{t h}, 5^{t h}, 6^{t h} ) |
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| 338 | The number of terms that are integers in the binomial expansion of ( (sqrt{7}+ ) ( sqrt[3]{5})^{35} ) is A . 4 B. 5 ( c cdot 6 ) D. |
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| 339 | If sum of the first 3 coefficients is 16 in the expansion ( left(x+frac{1}{x^{3}}right)^{n}, ) then find ( n ) A . 10 B. 8 ( c .5 ) D. |
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| 340 | The coefficient of ( x^{5} ) in the expansion of ( (x+3)^{8} ) is A . 1542 в. 1512 ( c .2512 ) D. 12 E . 4 |
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| 341 | WIL DULU – 20. T The remainder left out when out when 82n – (62)2n+1 is divided by 9 is: [2009] (2) 2 (6) 7 (c) 8 (d) oh |
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| 342 | The coefficients of XP and x9 in the expansion of (1+x)pta are [2002] (a) equal equal with opposite signs reciprocals of each other (d) none of these |
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| 343 | The term independent of ( x ) in the binomial expansion of ( left(1-frac{1}{x}+3 x^{5}right)left(2 x^{2}-frac{1}{x}right)^{8} ) ( mathbf{A} cdot-496 ) B . -400 c. 496 D. 400 |
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| 344 | If the second term in the expansion ( left[a^{frac{1}{13}}+frac{a}{sqrt{a^{-1}}}right]^{n} ) is ( 14 a^{5 / 2} ), then the value of ( frac{n C_{3}}{n C_{2}} ) is ( mathbf{A} cdot mathbf{4} ) B. 3 c. 12 D. 6 |
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| 345 | Find the middle term of the expansion of ( left(sqrt{boldsymbol{x}}-frac{mathbf{1}}{boldsymbol{x}}right)^{mathbf{6}} ) | 11 |
| 346 | Find the term independent of ( x ) in the expansion of ( left(1+x+2 x^{2}right)left(frac{3 x^{2}}{2}-frac{1}{3 x}right)^{9} ) |
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| 347 | In the expansion of ( (1+x)^{n} ) the coefficients of ( p^{t h} ) and ( (p+1)^{t h} ) terms are respectively ( p ) and ( q ) then ( p+q= ) ( mathbf{A} cdot boldsymbol{n} ) B. ( n+1 ) ( c cdot n+2 ) ( mathbf{D} cdot n+3 ) |
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| 348 | The coefficient of ( x ) in the expansion of ( left(1-x-x^{2}+x^{3}right)^{6} ) is ? ( A cdot 6 ) B. -6 c. -12 D. 12 |
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| 349 | The fourth term in the expansion of ( left(p x+frac{1}{x}right)^{n} ) is ( frac{5}{2} . ) Then This question has multiple correct options ( mathbf{A} cdot n=6 ) B . ( n=7 ) c. ( p=frac{1}{2} ) D. ( p=frac{1}{4} ) |
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| 350 | If ( A ) is the coefficient of the middle term in the expansion of ( (1+x)^{2 n} ) and ( B ) and ( mathrm{C} ) are the coefficients of two middle terms in the expansion of ( (1+x)^{2 n-1} ) then A. ( A+B=C ) в. ( B+C=A ) c. ( C+A=B ) D. ( A+B+C=0 ) |
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| 351 | The total number of rational terms in the expansion of ( (mathbf{7 3}+mathbf{1 1 9})^{mathbf{6 5 6 1}} ) A . 73 в. 729 ( c .728 ) D. 730 E. 732 |
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| 352 | The coeffecients of the middle term in the binomial expansion in powers of ( x ) of ( (1+alpha x)^{4} ) and ( (1+alpha x)^{6} ) is the same if ( boldsymbol{alpha} ) equals A ( cdot-frac{5}{3} ) в. ( frac{10}{3} ) ( c cdot frac{3}{10} ) D. |
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| 353 | The coefficient of ( x^{n} ) in the expansion of ( frac{1}{(1-x)(1-2 x)(1-3 x)} ) is A ( cdot frac{1}{2}left(2^{n+2}-3^{n+3}+1right) ) B. ( frac{1}{2}left(3^{n+2}-2^{n+3}+1right) ) c. ( frac{1}{2}left(2^{n+3}-3^{n+2}+1right) ) D. none of these |
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| 354 | The middle term in the expansion of ( (1+x)^{2 n} ) is A. ( frac{1.3 .5 ldots(2 n-1) 2^{n}}{n !} ) в. ( frac{1.2 .3 ldots(2 n-1) 2^{n} x^{n}}{n !} ) c. ( frac{1.3 .5 ldots(2 n-1) x^{n}}{n !} ) D. ( frac{1.3 .5 ldots .(2 n-1) 2^{n} x^{n}}{n !} ) |
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| 355 | Find the term of the expansion of ( (a+ ) ( b)^{50} ) which is the greatest in absolute value if ( |boldsymbol{a}|=sqrt{mathbf{3}}|boldsymbol{b}| ) |
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| 356 | 1 2 Murks) The sum of the rational terms in the expansion of (2 + 31/5,10 is. (1997 – 2 Marks) |
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| 357 | Fidn the ( 7^{t h} ) term from the end in the expansion of ( left(9 x-frac{1}{3 sqrt{x}}right)^{18}, x neq 0 ) |
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| 358 | 1. The larger of 9950 + 10050 and 10150 is (1982-2 Marks) 7 |
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| 359 | If ( (mathbf{1}+boldsymbol{x})^{mathbf{1 0}}=boldsymbol{a}_{mathbf{0}}+boldsymbol{a}_{mathbf{1}} boldsymbol{x}+boldsymbol{a}_{mathbf{2}} boldsymbol{x}^{mathbf{2}}+ ) ( ldots ldots a_{10} x^{10}, ) then value of ( left(a_{0}-a_{2}+a_{4}-a_{6}+a_{8}-a_{10}right)^{2}+ ) ( left(a_{1}-a_{3}+a_{5}-a_{7}+a_{9}right)^{2} ) is ( mathbf{A} cdot 2^{10} ) B . 2 ( c cdot 2^{20} ) D. None of these |
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| 360 | Find the coefficient of: ( x ) in the expansion of ( left(1-3 x+7 x^{2}right)(1-x)^{16} ) Enter 1 if answer is -19 otherwise enter 0 |
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| 361 | If the coefficients of ( x^{2} ) and ( x^{3} ) are both zero, in the expansion of the expression ( left(1+a x+b x^{2}right)(1-3 x)^{15} ) in powers of ( x ) then the ordered pair ( (a, b) ) is equal to: A. (28,315) B. (-54,315) c. (-21,714) D. (24,861) |
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| 362 | The number of integral terms in the expansion of ( (2 sqrt{5}+sqrt[66]{7})^{642} ) A. 105 B. 107 ( c .321 ) D. 108 |
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| 363 | If the number of terms in the expansion ( (2 x+y)^{n}-(2 x-y)^{n} ) is ( 8, ) then the value of ( n text { is } ldots ldots . . . . text { (where } n text { is odd }) ) A . 17 B. 19 c. 15 D. 13 |
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| 364 | ( (sqrt{2}+1)^{6}+(sqrt{2}-1)^{6}= ) A . 99 B. 98 c. 196 D. 198 |
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| 365 | Number of terms in the expansion of ( left(x^{1 / 3}+x^{2 / 5}right)^{40} ) with integral power of ( x ) is equal to |
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| 366 | Compute the summation ( sum_{k=0}^{27} kleft(_{k}^{27}right)left(frac{1}{2}right)^{k}left(frac{2}{3}right)^{27-k} ) | 11 |
| 367 | If the coefficient of the middle term in the expansion of ( (1+x)^{2 n+2} ) is ( p ) and the coefficients of middle terms in the expansion of ( (1+x)^{2 n+1} ) are ( q ) and ( r ) then A ( . p+q=r ) В. ( p+r=q ) ( mathbf{c} cdot p=q+r ) D. ( p+q+r=0 ) |
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| 368 | Expand the following binomial ( left(1+frac{x}{2}right)^{7} ) | 11 |
| 369 | Find the cube of the following binomial expressions: ( 4-frac{1}{3 x} ) |
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| 370 | If the middle term of ( (1+x)^{2 n} ) is the greatest term then ( x ) lies between A. ( n-1<x<n ) в. ( frac{n}{n+1}<x<frac{n+1}{n} ) c. ( n<x<n+1 ) D. ( frac{n+1}{n}<x<frac{n}{n+1} ) |
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